Selecting the number of principal components: estimation of the true rank of a noisy matrix

By Yunjin Choi, Jonathan E. Taylor, Robert Tibshirani. 2014

122 citations recorded by the source corpus.

Principal component analysis (PCA) is a well-known tool in multivariate statistics. One significant challenge in using PCA is the choice of the number of components. In order to address this challenge, we propose an exact distribution-based method for hypothesis testing and construction of confidence intervals for signals in a noisy matrix. Assuming Gaussian noise, we use the conditional distribution of the singular values of a Wishart matrix and derive exact hypothesis tests and confidence intervals for the true signals. Our paper is based on the approach of Taylor, Loftus and Tibshirani (2013) for testing the global null: we generalize it to test for any number of principal components, and derive an integrated version with greater power. In simulation studies we find that our proposed methods compare well to existing approaches.

Subfields: Analytical Chemistry, Statistics and Probability, Signal Processing

View Paper: https://www.semanticscholar.org/paper/49454e6f41ddb39e5f5cccb2006abdfcce99c2b1

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Citations in the corpus

Exact post-selection inference for the generalized lasso path (2018, background): “A number of extensions to diﬀerent problem settings are given in Lee and Taylor [23], Reid et al. [29], Loftus and Taylor [28], Choi et al. [8].”
Uniform asymptotic inference and the bootstrap after model selection (2015, methodology): “…an example of a broader framework that we might call selective pivotal inference, applicable in many other settings, as in, e.g., Taylor et al. (2016), Lee et al. (2016), Lee & Taylor (2014), Loftus & Taylor (2014), Reid et al. (2017), Choi et al. (2014), Fithian et al. (2014), Hyun et al. (2016).”
Statistical Learning with Sparsity: The Lasso and Generalizations (2015, methodology): “Sparse discriminant analysis from Fisher’s framework and optimal scoring were explored in Trendafilov and Jolliffe (2007), Leng (2008), Clemmensen et al. (2011), and Witten and Tibshirani (2011). Minorization algorithms are discussed in Lange, Hunter and Yang (2000), Lange (2004), and Hunter and Lan”
Bi-cross-validation for factor analysis (2015, methodology): “Similar to [33, 15] provides a sequential hypotheses testing method which is not based on asymptotics.”
Exact Post-Selection Inference for Sequential Regression Procedures (2014, methodology): “…work was completed, there was further progress on the application and development of exact post-selection inference tools, for example, by Lee and Taylor (2014), Reid, Taylor, and Tibshirani (2014), Loftus and Taylor (2014), Choi, Taylor, and Tibshirani (2014), and Fithian, Sun, and Taylor (2014).”
Selective Sequential Model Selection (2015, background): “…Loftus and Taylor (2014), who describe p-values for path algorithms that add groups of variables (instead of individual variables) at each step, and Choi et al. (2014), who describe p-values for steps of principle components analysis (each step marking the estimation of a principle component…”